Related papers: Arc-transitive pentavalent Cayley graphs with solu…
It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency.
We show that the directed labelled Cayley graphs coincide with the rooted deterministic vertex-transitive simple graphs. The Cayley graphs are also the strongly connected deterministic simple graphs of which all vertices have the same cycle…
A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It is known that the order $n$ and degree $d$ of a vertex-transitive nut graph satisfy…
A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and the Lie algebra Lie(G). A Cayley map can be thought of as a…
Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general…
A pair of graphs $(\Gamma,\Sigma)$ is said to be stable if the full automorphism group of $\Gamma\times\Sigma$ is isomorphic to the product of the full automorphism groups of $\Gamma$ and $\Sigma$ and unstable otherwise, where…
Given a finite group $G$ and a subset $X$ of $G$, the commuting graph of $G$ on $X$, denoted by ${\cal C}(G,X)$, is the graph that has $X$ as its vertex set with $x,y\in X$ joined by an edge whenever $x\neq y$ and $xy=yx$. Let $T$ be a…
A well-known conjecture of Babai states that if $G$ is a finite simple group and $X$ is a generating set of $G$, then the diameter of the Cayley graph $Cay(G,X)$ is bounded above by $(\log |G|)^c$ for some absolute constant $c$. The goal of…
In this work, we explore edge direction, transitivity, and connectedness of Cayley graphs of gyrogroups. More specifically, we find conditions for a Cayley graph of a gyrogroup to be undirected, transitive, and connected. We also show a…
A graph is edge-transitive if the natural action of its automorphism group on its edge set is transitive. An automorphism of a graph is semiregular if all of the orbits of the subgroup generated by this automorphism have the same length.…
This article lays the foundations for an analogue of geometric group theory that studies actions on graphs by right quasigroups, including racks and quandles. We study markings of graphs that realize racks, and we introduce (di)graph…
A connected, locally finite graph $\Gamma$ is a Cayley--Abels graph for a totally disconnected, locally compact group $G$ if $G$ acts vertex-transitively with compact, open vertex stabilizers on $\Gamma$. Define the minimal degree of $G$ as…
We prove the monotonic normalized heat diffusion property on distance-regular graphs with classical parameters of diameter $3$. Regev and Shinkar found a Cayley graph for which this property fails. On the other hand, this property has been…
A normally regular digraph with parameters $(v,k,\lambda,\mu)$ is a directed graph on $v$ vertices whose adjacency matrix $A$ satisfies the equation $AA^t=k I+\lambda (A+A^t)+\mu(J-I-A-A^t)$. This means that every vertex has out-degree $k$,…
A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e. a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra Lie(G). A prototypical example is the classical…
Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we…
Quasi-strongly regular graphs form a significant generalization of strongly regular graphs. We study the eigenvalues of a family of such graphs, $\Gamma_H(G)$, constructed from a finite group $G$ and a subgroup $H$. Our main results include…
Let $G$ be a finite group. The solubility graph associated with the finite group $G$, denoted by $\Gamma_{\cal S}(G)$, is a simple graph whose vertices are the non-trivial elements of $G$, and there is an edge between two distinct elements…
A subset $C$ of the vertex set of a graph $\Gamma$ is said to be $(\alpha,\beta)$-regular if $C$ induces an $\alpha$-regular subgraph and every vertex outside $C$ is adjacent to exactly $\beta$ vertices in $C$. In particular, if $C$ is an…
In this paper we study $G$-arc-transitive graphs $\Delta$ where the permutation group $G_x^{\Delta(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $\Delta(x)$ satisfies the two conditions given in the…